Einstein–Weyl structures on real hypersurfaces of complex two-plane Grassmannians

Xiaomin Chen Colloquium Mathematicum MSC: 53C40, 53C15. DOI: 10.4064/cm7922-8-2020 Published online: 15 March 2021

Abstract

We study real Hopf hypersurfaces with Einstein–Weyl structures in the complex two-plane Grassmannian $G_2(\mathbb {C}^{m+2})$, $m\geq 3$. First we prove that a real Hopf hypersurface with a closed Einstein–Weyl structure $W=(g,\theta )$ is of type (B) if $\nabla _\xi \theta =0$, where $\xi $ denotes the Reeb vector field of the hypersurface. Next, for a Hopf hypersurface with non-vanishing geodesic Reeb flow, we prove that there does not exist an Einstein–Weyl structure $W=(g,k\eta )$, where $k$ is a non-zero constant and $\eta $ is a one-form dual to $\xi $. Finally, it is proved that a real Hopf hypersurface with two closed Einstein–Weyl structures $W^\pm =(g,\pm \theta )$ is of type (A) or type (B).

Authors

  • Xiaomin ChenCollege of Science
    China University of Petroleum-Beijing
    Beijing, 102249, China
    e-mail

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